Even and Odd Functions


In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.


g(x) = x⁴ + 3x² − 1

Verified step by step guidance

1. To determine if a function is even, odd, or neither, we need to analyze the function's symmetry properties. A function g(x) is even if g(-x) = g(x) for all x in the domain, and it is odd if g(-x) = -g(x) for all x in the domain.
2. Start by substituting -x into the function g(x) = x⁴ + 3x² − 1 to find g(-x). This gives us g(-x) = (-x)⁴ + 3(-x)² − 1.
3. Simplify the expression for g(-x). Since (-x)⁴ = x⁴ and (-x)² = x², we have g(-x) = x⁴ + 3x² − 1.
4. Compare g(-x) with g(x). We find that g(-x) = x⁴ + 3x² − 1 is exactly the same as g(x) = x⁴ + 3x² − 1, which means g(-x) = g(x).
5. Since g(-x) = g(x), the function g(x) is even. Therefore, g(x) = x⁴ + 3x² − 1 is an even function.

Key Concepts

Even Functions

A function is classified as even if it satisfies the condition f(-x) = f(x) for all x in its domain. This means that the graph of an even function is symmetric with respect to the y-axis. Common examples include polynomial functions with only even powers of x, such as x² or x⁴.

Odd Functions

A function is considered odd if it meets the condition f(-x) = -f(x) for all x in its domain. This indicates that the graph of an odd function is symmetric with respect to the origin. Typical examples include polynomial functions with only odd powers of x, like x³ or x⁵.

Neither Even Nor Odd Functions

A function is classified as neither even nor odd if it does not satisfy the conditions for either classification. This can occur when a function contains both even and odd powers of x or does not exhibit symmetry about the y-axis or the origin. Analyzing the function's behavior at -x compared to x helps determine this classification.